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The derivative of the function/ sqrt(sin(5*x))

Derivative of sqrt(sin(5*x))

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  __________
\/ sin(5*x)
$$\sqrt{\sin{\left(5 x \right)}}$$ 
sqrt(sin(5*x))
Detail solution

  1. Let .

  2. Apply the power rule: goes to

  3. Then, apply the chain rule. Multiply by :

    1. Let .

    2. The derivative of sine is cosine:

    3. Then, apply the chain rule. Multiply by :

      1. The derivative of a constant times a function is the constant times the derivative of the function.

        1. Apply the power rule: goes to

        So, the result is:

      The result of the chain rule is:

    The result of the chain rule is:

The answer is:

The graph
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  5*cos(5*x)  
--------------
    __________
2*\/ sin(5*x)
$$\frac{5 \cos{\left(5 x \right)}}{2 \sqrt{\sin{\left(5 x \right)}}}$$
Simplify
The second derivative [src] 
    /                     2      \
    |    __________    cos (5*x) |
-25*|2*\/ sin(5*x)  + -----------|
    |                    3/2     |
    \                 sin   (5*x)/
----------------------------------
                4
$$- \frac{25 \left(2 \sqrt{\sin{\left(5 x \right)}} + \frac{\cos^{2}{\left(5 x \right)}}{\sin^{\frac{3}{2}}{\left(5 x \right)}}\right)}{4}$$
Simplify
The third derivative [src] 
    /         2     \         
    |    3*cos (5*x)|         
125*|2 + -----------|*cos(5*x)
    |        2      |         
    \     sin (5*x) /         
------------------------------
            __________        
        8*\/ sin(5*x)
$$\frac{125 \left(2 + \frac{3 \cos^{2}{\left(5 x \right)}}{\sin^{2}{\left(5 x \right)}}\right) \cos{\left(5 x \right)}}{8 \sqrt{\sin{\left(5 x \right)}}}$$
Simplify
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